Subnormal series - Knowlpedia

Let $G$ be a group. A subnormal series of $G$ is a finite chain

\{e\}=G_0 \lhd G_1 \lhd \cdots \lhd G_n = G

in which $G_{i-1}$ is a normal subgroup of $G_i$ for $1\le i\le n$ .

Examples

The chain $\{e\}\lhd G$ is a subnormal series for every group $G$ .
In $S_4$ , the chain $\{e\}\lhd V_4\lhd A_4\lhd S_4$ is a subnormal series.
The chain $\{e\}<\langle(12)\rangle<S_3$ is not subnormal because $\langle(12)\rangle$ is not normal in $S_3$ .

Remarks

Important examples include the derived series and the lower central series. A composition series is a subnormal series with simple successive factors.